Locally Most Powerful Invariant Test Research Articles

Correlation-Based Maximal Invariant Statistic for Spectrum Sensing

In this letter, spectrum sensing for wide-sense stationary signals with temporal correlation is studied. In order to perform uniformly most powerful invariant test (UMPIT) or locally most powerful invariant test (LMPIT), the ratio of the distributions of the maximum invariant statistics is derived in closed-form using Wijsman’s theorem. This ratio shows that UMPIT can be obtained when the signal-to-noise ratio (SNR) and the normalized auto-correlation matrix of the primary user signal are both known. In addition, the LMPIT with nominal SNR (LMPIT-NSNR) and the LMPIT with improved sample auto-correlation matrix (LMPIT-ISAC) are proposed based on this ratio. LMPIT-NSNR and LMPIT-ISAC are suitable for the case where the normalized auto-correlation matrix is known and the SNR is unknown, and for the case where both the normalized auto-correlation matrix and the SNR are unknown, respectively. Simulation results show that the performance of LMPIT-NSNR is robust to the fixed nominal SNR value and can approach the performance of UMPIT, and the LMPIT-ISAC performs better than the traditional covariance-based test.

CNN-aided multiple PU spectrum sensing in uncalibrated massive antennas SU system

This paper deploys the Convolutional Neural Network (CNN) to learn and set the statistical test in Spectrum Sensing (SS) task of multiple primary user (PU) sources in massive uncalibrated antennas of secondary users (SU) sharing the same spectrum resources. The proposed deep learning-based SS method (DL-SS) is based on the CNN architecture that has the capability of extracting features of the sample covariance matrices (SCMs) that are given as the network input, improving the overall performance and robustness. The proposed CNN-SS method is compared with nine recent multiple-antennas SS methods, namely the arithmetic–geometric detector (AGM), John’s detector (JD), sphericity detector (SD), generalized likelihood test (GLRT), locally most powerful invariant test (LMPIT), maximum–minimum eigenvalue detector (MME), covariance detector (CAV), Hadamard detector (HD) and volume detector (VD) methods; besides, the proposed method is also compared with five recent state-of-art CNN-based SS methodologies. Performance-complexity trade-off of the proposed and reference SS methods are corroborated via Monte Carlo Simulations (MCS). The proposed CNN-SS method under uncalibrated massive antennas reveals substantial benefits w.r.t. the reference methods and is competitive with others CNN-SS methodologies, both in terms of complexity and performance, achieving detection probability of Pd=0.9(@SNR=−20dB) under very low false alarm probability Pf=0.1. Under different figures of merit, the performance of the CNN-based SS detector has revealed to be indubitably superior regarding the state-of-art SS detectors. However, the proposed CNN-based SS detector presents relative computational complexity increases. Hence, to be effective, such a superior operational performance requires a very efficient processing structure in the SU base stations.

Testing Equality of Multiple Power Spectral Density Matrices

This paper studies the existence of optimal invariant detectors for determining whether $P$ multivariate processes have the same power spectral density. This problem finds application in multiple fields, including physical layer security and cognitive radio. For Gaussian observations, we prove that the optimal invariant detector, i.e., the uniformly most powerful invariant test, does not exist. Additionally, we consider the challenging case of close hypotheses, where we study the existence of the locally most powerful invariant test (LMPIT). The LMPIT is obtained in the closed form only for univariate signals. In the multivariate case, it is shown that the LMPIT does not exist. However, the corresponding proof naturally suggests an LMPIT-inspired detector that outperforms previously proposed detectors.

LMPIT-Inspired Tests for Detecting a Cyclostationary Signal in Noise With Spatio–Temporal Structure

In spectrum sensing for cognitive radio, the presence of a primary user can be detected by making use of the cyclostationarity property of digital communication signals. For the general scenario of a cyclostationary signal in temporally colored and spatially correlated noise, it has previously been shown that an asymptotic generalized likelihood ratio test (GLRT) and locally most powerful invariant test (LMPIT) exist. In this paper, we derive detectors for the presence of a cyclostationary signal in various scenarios with structured noise. In particular, we consider noise that is temporally white and/or spatially uncorrelated. Detectors that make use of this additional information about the noise process have enhanced performance. We have previously derived GLRTs for these specific scenarios; here, we examine the existence of LMPITs. We show that these exist only for detecting the presence of a cyclostationary signal in spatially uncorrelated noise. For white noise, an LMPIT does not exist. Instead, we propose tests that approximate the LMPIT, and they are shown to perform well in simulations. Finally, if the noise structure is not known in advance, we also present hypothesis tests using our framework.

Performance Analysis of Locally Most Powerful Invariant Test for Sphericity of Gaussian Vectors in Coherent MIMO Radar

Locally most powerful invariant test (LMPIT) for sphericity of Gaussian vectors has been derived by Ramirez et al. Nevertheless, the decision threshold of the LMPIT is not accurate and its detection performance has not yet been addressed. In this paper, the LMPIT is performed for target detection in multiple-input multiple-output (MIMO) radar, and its theoretical decision threshold as well as detection probability are accurately determined. Utilizing asymptotic expansion approach, we calculate the asymptotic null distribution as a function of central chi-square distributions, resulting in precise closed-form formula for thresholding. On the other hand, the nonnull distribution is approximated by weighted sum of noncentral chi-square distributions and Gamma distribution for close and far hypotheses, respectively. This enables us to derive a closed-form formula to precisely evaluate the detection power of the LMPIT. Numerical results demonstrate that our theoretical computations are very accurate in determining the decision threshold and predicting the behaviors of the LMPIT. Moreover, the superiority of the LMPIT for MIMO radar target detection over state-of-the-art methods is demonstrated for spatially colored but temporarily white noise.

Detection of Multivariate Cyclostationarity

This paper derives an asymptotic generalized likelihood ratio test (GLRT) and an asymptotic locally most powerful invariant test (LMPIT) for two hypothesis testing problems: 1) Is a vector-valued random process cyclostationary (CS) or is it wide-sense stationary (WSS)? 2) Is a vector-valued random process CS or is it nonstationary? Our approach uses the relationship between a scalar-valued CS time series and a vector-valued WSS time series for which the knowledge of the cycle period is required. This relationship allows us to formulate the problem as a test for the covariance structure of the observations. The covariance matrix of the observations has a block-Toeplitz structure for CS and WSS processes. By considering the asymptotic case where the covariance matrix becomes block-circulant we are able to derive its maximum likelihood (ML) estimate and thus an asymptotic GLRT. Moreover, using Wijsman's theorem, we also obtain an asymptotic LMPIT. These detectors may be expressed in terms of the Loeve spectrum, the cyclic spectrum, and the power spectral density, establishing how to fuse the information in these spectra for an asymptotic GLRT and LMPIT. This goes beyond the state-of-the-art, where it is common practice to build detectors of cyclostationarity from ad-hoc functions of these spectra.

Optimal invariant detection of a monochromatic plane wave with unknown amplitude, frequency, phase, direction of arrival and noise variance

This study deals with the problem of detecting a monochromatic plane wave with unknown amplitude, phase, temporal frequency and direction of arrival in complex white Gaussian noise with unknown variance. Depending on the natural invariance in scale, temporal modulation and spatial modulation of the problem, the uniformly most powerful invariant (UMPI) test is derived by using the statistical invariance principles. However, the UMPI test does not apply unless the signal-to-noise ratio (SNR) is known. However, it provides us with performance bound to evaluate any invariant test's performance when the SNR is unknown. Typically, the generalised likelihood ratio test (GLRT) and locally most powerful invariant (LMPI) test are derived as realisable suboptimal invariant tests with their performance comparison in different SNR through theoretical analysis. Computer simulation examples corroborate the authors analysis and indicate that the GLRT is close to the UMPI bound especially in the low probability-of-false-alarm (P FA) region of the receiver operating characteristic curve while the performance of the LMPI test is close to that of the UMPI test in the low SNR region.

On optimal tests for isotropy against the symmetric wrapped stable‐circular uniform mixture family

The family of Symmetric Wrapped Stable (SWS) distributions can be widely used for modelling circular data. Mixtures of Circular Uniform (CU) with the former also have applications as a larger family of circular distributions to incorporate possible outliers. Restricting ourselves to such a mixture, we derive the locally most powerful invariant (LMPI) test for the hypothesis of isotropy or randomness of directions-expressed in terms of the null value of the mixing proportion, p, in the model. Global monotonicity of the power function of the test is established. The test is also consistent. Power values of the test for some selected parameter combinations, obtained through simulation reveal quite encouraging performances even for moderate sample sizes. The P 3 approach (SenGupta, 1991; Pal & SenGupta, 2000) for unknown p and rho and the non-regular case of unknown a, the index parameter, are also discussed. A real-life example is presented to illustrate the inadequacy of the circular normal distribution as a circular model. This example is also used to demonstrate the applications of the LMPI test, optimal P 3 test and a Daviesmotivated test (Davies, 1977, 1987). Finally, a goodness-of-fit test performed on the data establishes the plausibility of the above SWS-CU mixture model for real-life problems encountered in practical situations.

Multivariate Tests with Incomplete Data

In the context of a normal model, testing problems with missing data are considered. Tests on means are treated when independent extra data on the first $p_1$ variates of $p$ variates is available in addition to complete data. For testing that the mean of the first $p_1$ variates is zero, the LRT is UMP invariant, but for testing that the whole mean is zero, no LMPI (locally most powerful invariant) test exists. Second, tests for independence are treated in similar situations, and an LMPI test for each situation is derived. In some situations it is found that the LRT for independence ignores the extra data.